A global root-finding method for high dimensional problems

TL;DR

This paper introduces a novel global root-finding method for high-dimensional problems that uses the center of mass of a specially constructed object, enabling efficient and exact solutions under broad conditions.

Contribution

It presents a new approach to solve f(x)=0 by computing the center of mass of an Omega-shaped object, which can be extended to multiple roots and implemented with adaptive Monte Carlo sampling.

Findings

Exact root position can be computed despite infinite mass.

Method converges exponentially fast with Monte Carlo sampling.

Applicable to high-dimensional problems with broad assumptions.

Abstract

A method to solve the problem f(x) = 0 efficiently on any n-dimensional domain Omega under very broad hypoteses is proposed. The position of the root of f, assumed unique, is found by computing the center of mass of an Omega-shaped object having a singular mass density. It is shown that although the mass of the object is infinite, the position of its center of mass can be computed exactly and corresponds to the solution of the problem. The exact analytical result is implemented numerically by means of an adaptive Monte Carlo sampling technique which provides an exponential rate of convergence. The method can be extended to functions with multiple roots, providing an efficient automated root finding algorithm.